Boro boast one of the tighest defences in the Championship and Gibson has been the key to that, having played every minute of league football for the club this season.
The club are seeking a return to the Premier League just one year after relegation, but Everton, West Ham and Leicester are all in line to offer Gibson a route back to the top flight even if Boro's promotion push falls at the final hurdle, according to Sky Sports.
Leicester have been keeping tabs on Gibson for several transfer windows and a defender remains their primary target.
However, Gibson signed a new five-year-deal at the Riverside earlier this season and Boro would demand £20m for their star defender to be prised away.
Gibson was the one bright spark in Middlesbrough's relegation season last year, earning a call up to the England senior team with his performances at the heart of the defence. He was also named Middlesbrough's player of the year.
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Gibson was born in a suburb of Middlesbrough and has been on Boro's books since 2005. He also has family connections to the club as the nephew of owner Steve Gibson.
The 25-year-old is unlikely to make a decision on his future until after the end of this season. Boro can guarantee a play-off place with victory over fellow promotion chasers Millwall on Saturday.
If Boro do not beat Millwall this weekend, the fight for play-off places will go down to the final weekend, when Tony Pulis' team travel to Ipswich.